

A1. Prove that (21n+4)/(14n+3) is irreducible for every natural number n.


A2. For what real values of x is √(x + √(2x1)) + √(x  √(2x1)) = A given (a) A = √2, (b) A = 1, (c) A = 2, where only nonnegative real numbers are allowed in square roots and the root always denotes the nonnegative root?


A3. Let a, b, c be real numbers. Given the equation for cos x:
a cos^{2}x + b cos x + c = 0,
form a quadratic equation in cos 2x whose roots are the same values of x. Compare the equations in cos x and cos 2x for a=4, b=2, c=1.


B1. Given the length AC, construct a triangle ABC with ∠ABC = 90^{o}, and the median BM satisfying BM^{2} = AB·BC.


B2. An arbitrary point M is taken in the interior of the segment AB. Squares AMCD and MBEF are constructed on the same side of AB. The circles circumscribed about these squares, with centers P and Q, intersect at M and N.
(a) prove that AF and BC intersect at N;
(b) prove that the lines MN pass through a fixed point S (independent of M);
(c) find the locus of the midpoints of the segments PQ as M varies.


B3. The planes P and Q are not parallel. The point A lies in P but not Q, and the point C lies in Q but not P. Construct points B in P and D in Q such that the quadrilateral ABCD satisfies the following conditions: (1) it lies in a plane, (2) the vertices are in the order A, B, C, D, (3) it is an isosceles trapezoid with AB parallel to CD (meaning that AD = BC, but AD is not parallel to BC unless it is a square), and (4) a circle can be inscribed in ABCD touching the sides.

